reserve x, x1, x2, y, y1, y2, z, z1, z2 for object, X, X1, X2 for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u9, u1, u2, v, v1, v2, w, w1, w2 for Element of E^omega;
reserve F, F1, F2 for Subset of E^omega;
reserve i, k, l, n for Nat;
reserve TS for non empty transition-system over F;
reserve s, s9, s1, s2, t, t1, t2 for Element of TS;
reserve S for Subset of TS;

theorem Th32:
  [[x1, x2], [y1, y2]] in ==>.-relation(TS) implies x1 in TS & y1
in TS & x2 in E^omega & y2 in E^omega & x1 in dom dom (the Tran of TS) & y1 in
  rng (the Tran of TS)
proof
  assume [[x1, x2], [y1, y2]] in ==>.-relation(TS);
  then x1, x2 ==>. y1, y2, TS by Def4;
  hence thesis by Th20;
end;
